Square-root higher-order Weyl semimetals

The mathematical foundation of quantum mechanics is built on linear algebra, while the application of nonlinear operators can lead to outstanding discoveries under some circumstances, such as the prediction of positron, a direct outcome of the Dirac equation which stems from the square-root of the Klein-Gordon equation. In this article, we propose a model of square-root higher-order Weyl semimetal (SHOWS) by inheriting features from its parent Hamiltonians. It is found that the SHOWS hosts both “Fermi-arc” surface and hinge states that respectively connect the projection of the Weyl points on the side surface and arris. We theoretically construct and experimentally observe the exotic SHOWS state in three-dimensional (3D) stacked electric circuits with honeycomb-kagome hybridizations and double-helix interlayer couplings. Our results open the door for realizing the square-root topology in 3D solid-state platforms.

where h H k = Φ † k Φ k and h K k = Φ k Φ † k represent the Hamiltonian of a stacked honeycomb sublattice and breathing kagome sublattice, respectively. Their explicit expressions are with h 11 = 3t 2 a , h 12 = t a t b + 2t a t z cos k z + (t a t b + 2t a t z cos k z )e −ik·a1 + (t a t b + 2t a t z cos k z )e −ik·a2 , h 22 = 3t 2 b + 12t b t z cos k z + 6t 2 z + 6t 2 z cos(2k z ), with We note that h H k and h K k have the same energy band solution, except that h K k has an additional flat band pinned to zero energy. The energy band solution of the h K k is E k = 0 and with t b = t b +2t z cos(k z ) and ∆(k) = (1+e ik·a1 +e ik·a2 )/3. The band structure of the original Hamiltonian is therefore given by ε k = ± √ E k .
Supplementary Figure 1. The admittance dispersion around the Weyl point K+ in the a qx − qy and b qx − qz planes. Here we are particularly interested in the 1st band, because the Weyl points appear at the intersection between the first and second energy bands. The spatial distribution of the Berry curvature for the 1st band around c kx − ky (kz = kzw) and d kx − kz (ky = 0) planes. Open and solid circles represent the Weyl points with opposite topological charges +1 and −1.

SUPPLEMENTARY NOTE 2. THE LINEAR ADMITTANCE SPECTRUM NEAR THE WEYL POINT AND THE BERRY CURVATURE
In this section, we demonstrate that the Weyl semimetal in our system hosts linear dispersion in all three dimensions in the vicinity of the Weyl points which act like monopoles of Berry curvature. To this end, we expand h H k in terms of Pauli matrix h H k = λ 0 σ 0 + λ x σ x + λ y σ y + λ z σ z with σ 0 the identity matrix, σ x , σ y and σ z being the Pauli matrices. The parameters λ i (i = 0, x, y, z) are explicitly expressed as Near the point K + = (4π/3, 0, k zw ), using the Taylor expansion, the parameters λ i (i = 0, x, y, z) of the effective Hamiltonian can written as: with q = k − K + . From Supplementary Eqs. (9), one can clearly see that the band linearly touches at K + , which is a typical feature of band crossing of Weyl semimetals. The energy bands around the Weyl point of SHOWS inherited from Supplementary Eqs. (9) are also linear. We then investigate the distribution of Berry curvature in momentum based on the low-energy effective Hamiltonian expanding around the Weyl points. It will be demonstrated that the Weyl points will generate Fermi arc states on the surface. We first consider the degenerate point at K + . Here, we plot the 3D band dispersion around K + in Supplementary Figures 1a, b. The band dispersion around K − is similar to the case around K + . Obviously, the band dispersion around the degenerate points along any direction is linear. Furthermore, the Berry curvature is expressed as where A µ = −i φ|∇ µ |φ is the berry connection, with µ = x, y, z and φ(q) being its wave function. Supplementary  Figures 1 c, d show that the flux of the Berry curvature flowing from K + to K − , which is similar to the magnetic monopole in momentum space. The monopole charge is defined as where FS is the curved surface surrounding the Weyl point. By evaluating C FS , we find that K + and K − are a pair of Weyl points with opposite charge +1 and −1, denoted by the open and solid circles respectively. This means that this 3D circuit system hosts four Weyl points that reside at the same admittance and is thus a Weyl semimetal.

SUPPLEMENTARY NOTE 3. THE 3D SQUARE-ROOT HOTI
The non-zero bulk polarization (in Supplementary Figure 2c) gives rise to the hinge states in a triangular prism sample with the dispersion connecting the projections of the Weyl points along the k z direction, as shown by the hinge state distribution in Supplementary Figure 2d.
It is worth mentioning that a 3D square-root HOTI can also emerge in our system for other parameters (see Supplementary Figures 2e-g). Comparing the bulk band structures with Supplementary Figure 2b, one can see that the band gap of high-order topological insulators always exists from K toK (see Supplementary Figure 2e ). In this region, the bulk polarization is always non-zero in Supplementary Figure 2f, but not the case for Supplementary  Figure 2c. For the surface band in Supplementary Figure 3a, the Zak phase is written as with u(k x , k z ) the wave functions of the surface states. We calculate the edge topological invariant [2,3] and find exactly the same phase transition point at ±k zw , as shown in Supplementary Figure 3b. Although it is not our purpose to resolve the debate between the edge invariant and bulk invariant to characterize the topology of hinge states, we sincerely show that these two approaches lead to the same prediction of the emergence of the hinge state in our lattice structure.

SUPPLEMENTARY NOTE 4. CALCULATIONS OF THE FERMI ARCS
The Fermi arc is the equi-energy contour of the surface states at a fixed j n = 0.004082 Ω −1 . Supplementary Figure  4a shows the Fermi arcs with the same energy of Weyl points. Because all the four Weyl points are at the same energy, the Fermi arcs connect two Weyl points with opposite charges. These surface states are clearly gapped, as shown in Supplementary Figures 4b-f. We analyze the Fermi arcs at f=835 kHz of Weyl points in frequency space. Supplementary Figure 5a shows the Fermi arcs with the same frequency of Weyl points. These surface states are clearly gapped, as shown in Supplementary Figures 5b-f.

SUPPLEMENTARY NOTE 5. MAPPING FROM KIRCHHOFF'S LAW TO SCHRÖDINGER EQUATION
We derive the relation between Kirchhoff's laws and Schroedinger equation, which enables us to calculate the frequency spectrum.
In electric circuits, the equation of motion is given by where V is the N -component voltage measured at each node against the ground and I is the N -component input current at each node. The homogeneous equations of motion (I = 0) can be rewritten as 2N differential equations of first order [1]: with ψ = (V(t), V(t)) T and the Hamiltonian block matrix being H S = i 0 C −1 L −1 0 . By diagonalizing H S , we can obtain the frequency dispersion ω(k x ).